In this talk, we will highlight recent developments in quaternionic analysis. We will introduce notions of continuous and holomorphic functions f: H -> H, where H denotes the skew-field (noncommutative field) of quaternions, for future use. In particular, an analogue of the Wiener algebra will be presented.

Obtaining a spectral representation for a bounded or unbounded normal operator on a quaternionic Hilbert space is an old problem with several claimed solutions dating back to the 1930s until the 1970s. Spectral representations for unbounded normal operators are motivated by a formulation of quantum mechanics over the quaternions which was first suggested by von Neumann and Birkhoff in 1936.

We will highlight issues with the approaches presented in these works and also present a spectral theorem for bounded and unbounded normal operators on a quaternionic Hilbert space which is based on the fairly recent notion of spherical spectrum. We shall see that in the quaternionic setting the spectral representation of a normal operators is given by

= \int_{C} Re(p) d + int_{C} Im(p) d,

where E is a projection-valued measure and J is an anti self-adjoint unitary operator that is an analogue of the imaginary unit in the classical case. Finally, we will use a recently obtained trigonometric moment characterization of sequences s: Z ->H to provide an alternative approach to obtaining a spectral theorem for unitary operators on a quaternionic Hilbert space.

This talk is based on joint works with Daniel Alpay, Fabrizio Colombo and Irene Sabadini.